Control segment-based lever-arm correction via curve fitting for high accuracy navigation

ABSTRACT

A method and apparatus for estimating a lever arm correction between the wide beam antenna and the spot beam antenna in a navigational satellite system is disclosed. A prescribed satellite maneuver is used to increase the accuracy of the predicted satellite attitude in determining the lever arm, and lever arm corrections are provided from a ground segment to navigation receivers using curve fit coefficients to permit long range estimates and to minimize data transmission requirements.

CROSS REFERENCE TO RELATED APPLICATIONS

This application is related to the following co-pending and commonly assigned patent application(s), all of which applications are incorporated by reference herein:

Application Ser. No. ______, entitled “SPACE-BASED LEVER ARM CORRECTION IN NAVIGATIONAL SYSTEMS EMPLOYING SPOT BEAMS,” by Jonathan A. Tekawy and Kevin M. O'Brien, and filed on same date herewith; and

Application Ser. No. ______, entitled “USER SEGMENT-BASED LEVER ARM CORRECTION VIA PRESCRIBED MANEUVER FOR HIGH-ACCURACY NAVIGATION,” by Jonathan A. Tekawy and Kevin M. O'Brien and filed on same date herewith.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates generally to space-based navigational systems, and in particular, to a control segment-based lever arm correction system applicable to navigational systems employing spot beams.

2. Description of the Related Art

The Global Positioning System (GPS) is a satellite system that transmits navigation signals that are received by ground-based GPS receivers and used to determine the position of the GPS receiver to a high degree of accuracy. GPS currently provides standard service to commercial receivers, and a higher accuracy service to military receivers authorized to receive such signals.

In current systems, the GPS navigation signal is transmitted via a wide beam satellite antenna disposed on each GPS satellite. The wide beam antenna permits any GPS receiver having a line-of-sight to the GPS satellite to receive the navigational signal, and when the navigation signal from a sufficient number of GPS satellites has been acquired, the GPS receiver can determine its position via a precision clock and well-known triangulation techniques.

Because GPS signals are also used in military applications, countermeasures can be expected to be applied in an attempt to reduce the effectiveness of the GPS system. One such countermeasure is jamming. To increase the effectiveness of the GPS signals in a jamming environment, a steerable high gain antenna may be used to transmit high intensity GPS signals via spot beams to areas where needed.

One difficulty with this approach is that the high gain spot beam antenna is typically physically displaced from the wide beam antenna, and consequently, the phase center of each antenna is also displaced as well. This displacement is known as the “lever arm” between the antennas, and left uncorrected, can negatively affect the ability of the GPS receivers to determine their position. Without any correction, the lever arm between the wide beam antenna and a 7 meter diameter spot beam antenna can contribute up to 4.4 meters of user range error (URE). Depending on the GPS satellite constellation, this uncompensated URE can produce up to nine meters (RMS) of vertical (altitude) navigation error, which is a factor of 10 higher than the performance of the current GPS constellation of 0.9 meters (RMS). In civil aviation applications, such errors are sufficient to result in loss of life, and in military applications, they can result in increased collateral damage, and increased sortie and weapon consumption to perform the same mission. Further exacerbating this problem is the fact that in order to maintain proper Sun and Earth pointing, the GPS satellites are required to perform attitude maneuvers. Such maneuvers can be very large, particularly about yaw axis.

To achieve such high-accuracy navigation demanded for many missions, the GPS system must provide users with real-time corrections of the spot beam antenna phase center location relative to the Earth coverage antenna phase center, even while the spot beam antenna is moving to track specific terrestrial locations.

What is needed is an apparatus and method for computing a correction for lever arm related errors, and for incorporating this correction in navigation computations. The present invention satisfies these needs.

SUMMARY OF THE INVENTION

To address the requirements described above, the present invention discloses a method and apparatus for estimating a lever arm correction between a wide beam antenna and a spot beam antenna. The method comprises the steps of generating a predicted satellite attitude over the time period T from a predicted position of the at least one satellite over the time period T and a profile of a prescribed satellite attitude maneuver over the time period T; generating predicted gimbal angles of the spot beam antenna over the time period T from the predicted satellite attitude over the time period T and the terrestrial location of the of the spot beam on Earth when illuminating a navigational receiver over the time period T; predicting the lever arm correction between the wide beam antenna and the spot beam antenna over the time period T from the predicted gimbal angles of the spot beam antenna over the time period T and the predicted satellite attitude over the time period T; and transmitting the predicted lever arm correction from the command station to the navigational receiver. The apparatus comprises a lever arm correction module, for predicting a lever arm correction between the first satellite antenna and the second satellite antenna, and a transmitter for transmitting the predicted lever arm correction to a navigational receiver via the navigational satellite. In one embodiment, the lever arm correction module comprises a satellite position predictor module, for generating a predicted navigational satellite bus position over the time period T, a satellite attitude predictor module, for generating a predicted navigational satellite bus attitude over the time period T from the predicted navigational satellite bus position over the time period T and a profile of a prescribed navigational satellite bus attitude maneuver over the time period T, and a lever arm prediction module, for predicting the lever arm correction between the first antenna and the second antenna over the time period T from a terrestrial location of the spot beam illuminating the navigational receiver over the time period T, the predicted navigational satellite bus attitude over the time period T, and a geometry between the navigational satellite bus and the second satellite antenna.

BRIEF DESCRIPTION OF THE DRAWINGS

Referring now to the drawings in which like reference numbers represent corresponding parts throughout:

FIG. 1A is an illustration of a three-axis stabilized satellite;

FIG. 1B is an illustration of an embodiment of a gimbal assembly;

FIG. 2 is a diagram depicting a functional architecture of a representative satellite attitude control system;

FIG. 3 is a diagram depicting geometrical relationships between the satellite, the Earth and the Sun;

FIGS. 4A-4E are diagrams depicting the yaw noon maneuver at orbital midnight for a zero Sun angle;

FIG. 5 is a diagram depicting the yaw attitude angle as a function of the orbit angle and the Sun angle;

FIG. 6 is a diagram illustrating an exemplary embodiment of a technique for estimating the lever arm correction between the wide coverage antenna and the spot beam antenna;

FIG. 7 is a diagram illustrating architectural elements of a space-based navigational system;

FIGS. 8A and 8B are diagrams illustrating exemplary process steps that can be used to implement one embodiment of the invention, and how these process steps can be allocated to the architectural elements;

FIG. 9 is a plot of the yaw attitude angle of the satellite 100 as a function of the orbit angle and the Sun angle for an exemplary prescribed satellite 100 maneuver profile;

FIG. 10 is a diagram showing typical lever arm motion over time; and

FIG. 11 is a diagram illustrating one technique that can be used to arrive at the prescribed maneuver profile.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS

In the following description, reference is made to the accompanying drawings which form a part hereof, and which is shown, by way of illustration, several embodiments of the present invention. It is understood that other embodiments may be utilized and structural changes may be made without departing from the scope of the present invention.

FIG. 1A illustrates a three-axis stabilized satellite or spacecraft 100. The satellite 100 has a main body 102 (which may be referred to as the “satellite bus”) one or more solar panels 104, one or more navigation beam antennas 106E and 106S, and a telemetry and command antenna 108 which is used to communicate with a control ground station. The satellite 100 may also include one or more sensors 110 to measure the attitude of the satellite 100. These sensors may include sun sensors, earth sensors, and star sensors. Since the solar panels are often referred to by the designations “North” and “South”, the solar panels in FIG. 1A are referred to by the numerals 104N and 104S for the “North” and “South” solar panels, respectively.

The three axes of the spacecraft 100 are shown in FIG. 1A. The pitch axis lies along the line of the solar panels' 140N and 140S mutual rotation axes. The roll and yaw axes are perpendicular to the pitch axis and lie in the directions and planes shown.

In the illustrated embodiment, the satellite 100 includes a first navigation beam antenna 106E and a second navigation beam antenna 106S. The first navigation beam antenna 106E is a wide-beam antenna which transmits a navigation signal with a beamwidth covering the widest range of the Earth's surface possible from that satellite's altitude at any time, and is directed toward the Earth along the yaw axis. Since this antenna 106E offers the widest coverage of the Earth's surface, it is typically not steerable. The navigation system uses a constellation of such satellites 100 to provide coverage anywhere on the Earth's surface by at least 4different satellites at all times, thus permitting the navigational signal transmitted by the satellites to be used to determine the location and clock bias of the receiver using triangulation techniques.

The second navigation beam antenna 106S is a steerable spot beam antenna that provides a second navigation signal in a much narrower navigation beam. This allows transmission of a higher-strength beam to selected points on the ground without requiring excessive transmitter power, thus reducing the effectiveness of countermeasures such as jamming. Since the required service area includes substantially the entire surface of the Earth and the beamwidth of the spot beam antenna 106S is not wide enough to cover the entire surface area, the boresight of the spot beam antenna 106S can be steered about to direct the spot beam where desired. Such steering can be accomplished mechanically, by use of a gimbal structure driven by gimbal motors, or electronically, using phased arrays, for example.

FIG. 1A also shows the phase center 112E of the wide beam antenna 106E and the phase center 112S of the spot beam antenna 106S. Since the spot beam antenna 106S is offset from the wide beam antenna 106E, the phase centers 112S, 112E of the antennas are separated by an antenna lever arm 114, which is represented as a vector originating at the phase center 112E of the wide beam antenna 106E and extending to the phase center 112S of the spot beam antenna 106S. Due to motion of the satellite bus 102 and the spot beam antenna 106S and other factors, the antenna lever arm 114 does not remain fixed, but can vary substantially over time. This variance is enough to add a significant uncertainty in the ability of a navigation receiver (such as a GPS receiver) to determine its location when a navigation signal is received via the spot beam.

The spot beam antenna 106S may be steered electronically (by appropriate phasing of elements in a scanning array) or mechanically (by use of a non-scanning antenna and a gimbal assembly), or a combination of both. FIG. 1B illustrates an embodiment using a gimbal assembly 152 having an inner gimbal 154A and an outer gimbal 154B. The inner gimbal 154A is associated with inner gimbal coordinate frame 156B while the outer gimbal 154B is associated with an outer gimbal coordinate frame 156B. Driven by gimbal motors or other devices (not illustrated), the inner and outer gimbals 154A, 154B angularly direct the antenna 106S boresight (which is represented in antenna signal boresight coordinate frame 156D) where desired to transmit the navigation signal spot beam using the antenna 106S. Both inner and outer gimbals 154 typically include a potentiometer or other means to measure the gimbal angle.

To determine the angle at which the inner and outer gimbals 154 should be positioned to direct the spot beam antenna 106S to the desired scan locations (to perform a specific mission profile), the spacecraft 100 determines its attitude via the attitude sensors 110, which may be mounted on the satellite bus or body 102. Using the measured satellite attitude and the angular and translational displacement between the satellite body 102 and the spot beam antenna 106S (as expressed by coordinate systems 156A-156D), the satellite 100 can determine the appropriate gimbal angles, and commands the gimbal motors to move the gimbals 154 to the appropriate positions.

FIG. 2 is a diagram depicting the functional architecture of a representative attitude control system 200. Control of the spacecraft is provided by a computer or spacecraft control processor (SCP) 202. The SCP 202 performs a number of functions which may include post ejection sequencing, transfer orbit processing, acquisition control, station-keeping control, normal mode control, mechanisms control systems support, among others.

The SCP 202 may implement one or more processing modules such as antenna control module 276, which is used to control the satellite spot beam antenna drive 274 to slew the spot beam antenna 106S to the appropriate orientation and to transmit a navigation signal. Alternatively, the antenna control module 276 can be implemented in a different processor or in dedicated circuitry.

Input to the spacecraft control processor 202 may come from any combination of a number of spacecraft components and subsystems, such as a transfer orbit sun sensor 204,an acquisition sun sensor 206, an inertial reference un 210, an operational orbit Earth sensor 212, a normal mode wide angle sun sensor 214, a magnetometer 216, and one or more star sensors 218.M

The SCP 202 generates control signal commands 220 which are directed to a command decoder unit 222. The command decoder unit operates the load shedding and battery charging systems 224. The command decoder unit also sends signals to the magnetic torque control unit (MTCU) 226 and the torque coil 228.M

Wheel torque commands 262 are generated by the SCP 202 and are communicated to the wheel drive electronics 240 which command the speed of the reaction wheels in reaction wheel assembly(s) 244. Typically, the spacecraft 100 includes four reaction wheels, at least one in each orthogonal direction, and one for redundancy purposes. The speed of the reaction wheels is also measured and fed back to the SCP 202 by feedback control signal 264. The SCP 202 also communicates commands and data 254 with command stations (further described in connection with FIG. 7) via a satellite transmitter/receiver (or transceiver) 258.M

The foregoing describes an exemplary space stabilized satellite attitude control system. The present invention can be implemented with other attitude control system designs as well.

The SCP 202 may include or have access to memory 270, such as a random access memory (RAM). Generally, the SCP 202 operates under control of an operating system 272stored in the memory 270, and interfaces with the other s and generate outputs, including commands. Applications running in the SCP 202 access and manipulate data stored in the memory 270. The spacecraft 100 may also comprise an external communication device such as a satellite link for communicating with other computers at, for example, a ground station. If necessary, operation instructions for new applications can be uploaded from ground stations.

In one embodiment, instructions implementing the operating system 272, application programs, and other modules are tangibly embodied in a computer-readable medium, e.g., data storage device, which could include a RAM, EEPROM, or other memory device. Further, the operating system 272 and the computer program are comprised of instructions which, when read and executed by the SCP 202, cause the spacecraft processor 202 to perform the steps necessary to implement and/or use the present invention. Computer program and/or operating instructions may also be tangibly embodied in memory 270and/or data communications devices (e.g. other devices in the the invention. As such, the terms “program storage device,” “article of manufacture” and “computer program product” as used herein are intended to encompass a computer program accessible from any computer readable device or media.

As described above, it is desirable to correct for navigation errors caused by the lever arm 114 between the wide beam antenna and the steerable spot beam antenna. One approach taken to correct for such errors is to compute the antenna lever arm 114 in the satellite coordinate frame, and transmit this information to the GPS receivers in the navigational message. The GPS receiver then uses this information to compute corrections in the GPS (e.g. Earth Centered Earth Fixed) coordinate frame. However, this technique is complicated by the maneuvers that the satellite 100 must undergo to direct the solar panels 104 at the Sun and the wide beam antenna 106E at the Earth. Depending on the Sun angle, this requires a rather substantial yaw maneuver twice each orbit (at orbit noon and midnight).

FIG. 3 is a diagram depicting the geometrical relationships between the satellite 100,the Earth 308 and the Sun 306. Each satellite 100 of t around the Earth 308, thus defining an orbital plane 304. The angle 0 between the orbital plane 304 and the Sun 306 is the Sun angle. The satellite 100 orbits the Earth 308, always directing the antenna 106E at the Earth 308. The satellite 100 attitude and solar panels 104are adjusted to keep the solar panels 104 directed perform a large rotation about the yaw axis at the orbital midnight (0 degree orbit angle) and the orbital noon (180 degree orbit angle). The smaller the Sun angle θ, the larger the required rotation about the yaw axis.

FIGS. 4A-4E are diagrams depicting the yaw maneuver at orbital midnight for a zero Sun angle (wherein the Sun 306 is in the orbital plane 304). FIG. 4A illustrates the satellite 100 attitude and solar panel 104 orientation as the orbital midnight is approached. Note that the solar panels 104 are canted at an angle to direct them to be perpendicular to the Sun's rays. Shortly before the Sun 306 begins to be eclipsed by the Earth 308, the satellite 100M begins its maneuver and yaws counter clockwise as shown in FIG. 4B through FIG. 4D. When the Sun 306 fully emerges from the eclipse, the satellite 100 has yawed 180 degrees from its initial position.

FIG. 5 is a diagram plotting the yaw angle as a function of the orbit angle. At higher Sun 306 angles θ, the yaw angle movement is gradual, as shown in trace 502. However, at low Sun 306 angles θ (e.g. Sun angles less than about 5 degrees), the yaw angle abruptly transitions 180 degrees at orbital angles of 0 and 180 degrees, with resulting high yaw angular rates, as shown in trace 504. Such low Sun angle conditions occur over a significant portion of the year.

FIG. 6 is a diagram illustrating an exemplary embodiment of a technique for estimating the lever arm 114 correction between the wide coverage antenna 106E and the spot beam antenna 106S. In block 602, a prediction of the satellite 100 position over a time period T is generated. Block 604 illustrates the generation of a prediction of the satellite 100attitude profile over the time period T from am coverage antenna and the spot beam antenna over the time period T using the profile of the predicted satellite attitude maneuver over the time period T and a position of the terrestrial location of the spot beam on the Earth (e.g. when illuminating one or more of the navigational receivers).

FIG. 7 is a diagram illustrating architectural elements of a space-based navigational system 700. The navigational system 700 comprises a space segment 704 having a plurality of navigational satellites 100 (hereinafter alternatively referred to as spacecraft) in orbit around the Earth. Each of the satellites 100 includes a first transmitter 710, which transmits a first navigational signal using the wide beam antenna 106E and a second transmitter 712,which, using the spot beam antenna 106S, transmits an user segment 706. The spot beam 719 covers an area 715 on the Earth's surface, and is centered on the target spot beam position 717. Both the first navigational signal and the second navigational signal include information that can be used to determine the location of the navigational receiver 716. In a jamming environment, a navigation receiver may lose lock on the first navigation signal and will not be able to produce any positional solution. The second navigational signal, however, is typically of greater power by virtue of being transmitted via a spot beam, and, will be used for positional determination using the navigational receiver 716 in the jamming environment. The satellites 100 in the space segment 704 are controlled via commands from the ground segment 702, which comprises one or more ground-based command stations 707, each including a command center 708and a communicatively coupled antenna 714 for transmitting an commands to and from the satellites 100 of the space segment 704.

The operations illustrated in FIG. 6 can be performed on-board the satellite ground segment 702 controlling the satellite 100, by the navigational receivers 716 making up the user segment 706, or any combination thereof.

However, while a fixed antenna lever arm 114 can be computed in the satellite coordinate frame 156 and transmitted to the GPS receiver in the navigational message, this technique has serious disadvantages. First, this approach also cannot be applied to GPS systems that use steerable spot beam antennas, because the GPS receivers do not know the satellite 100 attitude, very accurately, and thus have insufficient information to compute an accurate estimate of the lever arm 114 for all extended times. Second, even if the GPS receivers were provided with the satellite 100 attitude, this technique requires that the GPS receivers perform the majority of the correction processing. Since the GPS receivers are often required to be very small, lightweight, and low power consumption devices, such processing can be a significant problem. Also, because of satellite attitude control system 200 limitations, the high yaw rate maneuvers that occur at low Sun 306 angles (such as shown in FIG. 5) add substantial uncertainty to the antenna lever arm 114 corrections transmitted to the GPS receivers.

One solution to these problems is to compute discrete lever arm 114 corrections in the GPS coordinate frame in the satellite 100 or in the ground segment 702 and periodically transmit the corrections to the GPS receivers. Since the correction data is in the GPS coordinate frame, it can be applied to the satellite orbital motion data already provided by the GPS system, and thus requires only minimal GPS receiver processing. However, such discrete corrections do not provide solutions in between the discrete points, and adding additional discrete points would consume a prohibitively large portion of the transmission bandwidth in the navigation message.

Finally, a problem with either of the foregoing correction approaches is that the corrections must be extrapolated for some period into the future. The first approach (that of transmitting the fixed antenna lever arm position in the satellite 100 coordinate frame in the navigational message and relying upon the GPS receivers to use this information along with a satellite yaw steering maneuver to enable computation of the corrections in a GPS coordinate frame), relies on an idealized (and often inaccurate) prediction of the satellite 100 attitude for all relevant times. The second app into the future, and during those 15 minutes, without additional information or constraints, this solution can yield large errors.

FIGS. 8A and 8B are diagrams illustrating exemplary process steps that can be used to implement one embodiment of the invention, and how these process steps can be allocated to the architectural elements (e.g. the control segment 702, the space segment 704, Mand the user segment 706).

Referring first to FIG. 8A, the predicted position of the satellite 100 over a time period T is computed, as shown in block 802. The predicted position of the satellite 100 can be determined using a variety of methods well known in the art, and can include parameters to model satellite maneuvers (whether for mission specific or station keeping purposes).

Next, as shown in block 806, a prediction of the satellite's attitude over the same time period T is computed. This is accomplished using the computed satellite position (from the computation shown in block 802), and a prescribed satellite maneuver 808. The prescribed satellite attitude maneuver 808 is performed to direct the yaw axis of the satellite bus 102 at the Earth 308 while directing the satellite solar panels 104 at the Sun 306, as illustrated in FIGS. 4A-4E, while requiring satellite angular rates and/or accelerations only within the satellite's capabilities. Typically, the maneuver is required about only the yaw axis.

FIG. 9 is a plot of the yaw attitude angle of the satellite 100 as a function of the orbit angle 310 and the Sun 306 angle for an exemplary profile of a prescribed satellite 100maneuver. The prescribed maneuver can be defined such than capabilities and limitations of the satellite's attitude control system 200, the satellite 100 can execute the prescribed attitude maneuver within a specific angular tolerance, the selection of such tolerance is a function of the desired lever arm estimation accuracy. In one embodiment, the value of this tolerance, expressed as the difference between the predicted satellite attitude and the actual satellite attitude is less than 0.2 degrees.

As was true in FIG. 5, trace 502 illustrates yaw angle profile for a high Sun 306 angle θ. At high Sun 306 angles θ, the required yaw rates and accelerations are relatively small, so that ideal and prescribed yaw profiles are the same. However, at low Sun 306 angles θ (e.g. less than about 5 degrees), the required yaw rates and accelerations are much higher for the ideal profile, as shown in trace 504. These yaw rates are high enough so that the satellite's attitude control system 200 cannot follow the ideal profile, and thus, the deviation from the expected satellite 100 attitude angle and the actual satellite 100 attitude angle is significant enough to negatively impact the estimation of the lever arm 114. Also, the maneuver is initiated based upon sensor (e.g. Sun, Earth, and/or star sensors) measurements, hence the time the maneuver is initiated is another source of unpredictability. To prevent this, the satellite is commanded to perform a prescribed satellite maneuver 808.

This prescribed satellite maneuver 808 can be defined such that, when can execute the prescribed attitude maneuver to within specified tolerances. The maneuver profile can be arrived at by (1) limiting the satellite attitude angular rate and/or acceleration in all axes (e.g. pitch, roll, and yaw) or a single axis (e.g. yaw only) to a maximum value, (2) limiting the error between the predicted satellite attitude and the actual satellite attitude (when attitude control system 200 limitations are considered), (3) and/or placing other suitable restrictions upon the commanded satellite motion. For example, the prescribed satellite maneuver 808 can be defined so as to limit the error between the predicted satellite attitude and the actual satellite attitude to less than 0.2 degrees and/or limit the yaw attitude rate to a particular value.

An exemplary profile of a prescribed satellite maneuver is illustrated in FIG. 9 as trace 902. Note that the difference between the profile of the ideal motion (trace 504) and the profile of the prescribed maneuver (trace 902) is small enough so that the solar panels 104 remain adequately directed at the Sun, yet by commanding the satellite attitude control system 200 to perform the prescribed maneuver instead of the ideal maneuver, the resulting actual satellite 100 attitude becomes predictable enough to allow the lever arm to be accurately predicted.

Returning to FIG. 8A, spot beam gimbal angles are computed, as shown in block 816. This is accomplished using the predicted satellite 100 attitude over the time period T (computed in block 806), the gimbal geometry 810, and the target position of the spot beam (e.g. where the spot beam is aimed to illuminate one or more terrestrial targets such as navigational receivers) over the time period T , as computed in block 814. The position of the target spot beam over time period T may be computed from a mission profile 812, which is the mission that one or more of the satellites 100 in the navigation system 700 is to perform (for example, which areas on the Earth's surface the spot beam antenna will be directed at).

The gimbal geometry 810 (which includes, for example, a description of the dimensions and orientation of the gimbal structures and joints used to rotate the gimbals to direct the spot beam antenna 106S or equations describing a relationship between pointing directions and gimbal angles) can be loaded into a memory of the satellites 100 of the space segment 704, or can be uplinked to the space segment 704 from the ground segment, if desired. The geometry can include, for example, positional vectors which together extend from the wide coverage antenna 106E phase center 112E to the spot beam antenna 106S and its phase center 112S. This may include, for example, a vector between the phase center 112E of the wide coverage antenna 106E to the satellite bus 102, from the satellite bus 102to the rotational axis of the spot beam antenna 106S point to the rotational axis of the outer spot beam antenna 106S gimbal motor 156C, and from the second spot beam antenna 106S gimbal motor to the spot beam antenna 106 phase center. This geometry may also be expressed in terms of coordinate transformations. For example, from the wide-coverage antenna 106E phase center 112E to the satellite bus coordinate frame 156A, through a coordinate transformation from the satellite bus coordinate frame 156A to the inner gimbal coordinate frame 156B, through an additional coordinate transformation from the inner gimbal coordinate frame 156B to the outer gimbal coordinate frame 156C and finally through a coordinate transformation from the outer gimbal coordinate frame 156C to the spot beam antenna coordinate frame 156D.

Turning to FIG. 8B, the spot beam gimbal angles are used to predict the lever arm 114 vector. In one embodiment, the lever arm vector is predicted in the satellite bus coordinate system 156A. Via appropriate coordinate transformations, this predicted lever arm vector 114 can be transformed from any of the foregoing coordinate systems (including the satellite bus 156A and the spot beam antenna 156D) to a desired coordinate frame. In one embodiment, the predicted lever arm vector 114 is transformed from the satellite bus coordinate frame 156A into an Earth Centered Inertial or Earth Centered Earth Fixed (ECEF) coordinate frame so as to permit the navigational receiver 716 to apply the lever arm correction to its derived position estimates with a minimum of computations. This embodiment is shown in FIG. 8B.

In block 828, the lever arm 114 vector LAs_(SV)(x(t),y(t),z(t)) is transformed into Earth-Centered-Earth-Fixed (ECEF) coordinates so that this information may be used by the navigational receivers 716 without further conversion. The conversion can be performed with an estimate of the satellite's attitude computed in block 806, or can use the actual measured satellite attitude, transmitted from the satellite 100 to the ground segment 702.M

FIG. 10 is a diagram illustrating the behavior of the lever arm expressed in component parts LA_(ECEF)(x(t)), LA_(ECEF)(y(t)), LA_(ECEF)(z(t)). Because the lever arm 114vector is time dependent, the transmits would consume a significant amount of available transmission bandwidth. In a preferred embodiment of the invention, this difficulty is overcome by using curve-fitting techniques (including, for example linear and non-linear regression) to identify closed form equations that describe the behavior of the lever arm over time during the time period T. In particular, the time-varying lever arm vector is expressed into its component parts (e.g. x(t), y(t), and z(t)), and each time-varying component part is curve-fit to an equation definable by combination of functions, each defined by one or more functional coefficients. This is shown in block 824.M

For example, each time-varying component can be fit to an n^(th) A order polynomial ${x(t)} = {{\sum\limits_{i = 1}^{n}{a_{i}{x^{i}(t)}}} = {a_{0} + {a_{1}{x(t)}} + {a_{2}{x^{2}(t)}} + \ldots + {a_{n}{x^{n}(t)}}}}$

The appropriate curve-fitting technique (e.g. the characteristics and order of the equation) can be selected according to the characteristics of the component part of the time varying lever arm 114 vector. For example, after z(t) for the time period T has been computed, z(t) can be evaluated to determine whether it is representable more accurately with a second, third, or fourth order polynomial. This can be accomplished, for example, by computing the mean squared error for each representation, and selecting the characteristics of the curve fit to minimize mean squared error. It may also be desirable to use different curve-fitting techniques and characteristics for the different component parts of the lever arm 114 vector.

Next, the lever arm vector coefficients are sent to the satellite 100, as shown in block 832. The satellite receives the vector, as shown in block 833, and transmits the lever arm vector coefficients to the navigational receivers 716, as shown in block 834. The navigation receivers receive the coefficients, convert those coefficients to a time-varying lever arm correction prediction in ECEF coordinates, and apply the lever arm corrections at the appropriate time, as shown in blocks 835-838.M

The operations illustrated blocks 802, 806, 814, 816, 820, 822, 824, and 832 of FIGS. 8A and 8B as being performed by the ground segment 702 can be implemented by software modules executed by a processor resident in the command station 707, software modules executed by special or general purpose auxiliary processors in the command station 707according to instructions stored in a memory communicative by dedicated electronic circuitry in the command station. For example, in one embodiment, the operations shown in blocks 802, 806, 814, 816, 820, 822, and 824 are performed by a lever arm correction module 801, which can be implemented in the command station processor 718, an auxiliary processor, or dedicated electronic circuitry. Further, the lever arm correction module 801 can include a satellite position predictor module 804, a satellite attitude predictor module 806, and a lever arm prediction module 826, which may further comprise a coordinate transformer 828 and curve fitting module 830 to perform the associated tasks shown in FIGS. 8A and 8B. These modules may also be implemented in the command station processor 718, an auxiliary processor, or dedicated electronic circuitry.

Generation of Prescribed Maneuver Profile

FIG. 11 is a diagram illustrating one technique that can be used to arrive at the prescribed maneuver profile. In block 1102, the performance of the satellite attitude control system (ACS) is determined, using specifications and/or test data of the applicable components, including, for example, the reaction or momentum wheels, ACS thrusters, and attitude sensors. In block 1104, this data is used to determine the maximum angular rate and/or acceleration that can be achieved by the spacecraft 100 under the most demanding maneuvers it will be required to perform, typically the yaw maneuver at orbital noon and midnight for low Sun angles, as depicted in FIGS. 4A-4E. Since the yaw channel is typically most critical, a yaw steering profile can be examined. This can be accomplished using digital, analog, or hybrid control simulations of the satellite ACS. In block 1106, an ideal attitude profile is generated. This profile does not consider the limitations of the ACS, and yields zero pointing errors about all axes and zero pointing errors of the solar panels at all times. In block 1108, a candidate prescribed attitude profile is selected. The candidate profile is selected so as to attempt to match or closely approximate the ideal profile generated in block 1106. Preferably, the candidate attitude profile is computed using closed form equations with parametric variability. The candidate profile will generally deviate from the ideal profile by an error, and the goal is to minimize this error.

Using this information, the candidate prescribed attitude profile is tested to determine if the spacecraft 100 can perform the maneuver described in the profile with sufficient (and typically nearly perfect) accuracy. Again, this can be performed with digital, analog, or hybrid simulations. If the test indicates that the satellite 100 can follow the prescribed maneuver profile, even better performance can be achieved by selecting a more demanding prescribed maneuver profile (using perhaps another candidate profile that follows the ideal profile more closely) that yields smaller errors when compared to the ideal profile, processing can return to block 1106 for the generation and eventual evaluation of that profile. If the test indicates that the satellite 100 cannot follow the prescribed maneuver profile, then the candidate prescribed maneuver profile is too demanding. The profile is adjusted in block 1114 and re-evaluated in steps 1108-1110. The foregoing process is iteratively repeated until a prescribed maneuver profile is found that achieves the least error between itself and the ideal maneuver profile, while still allowing the spacecraft to follow it essentially error free.

An example candidate prescribed yaw attitude profile is given below ψ_(p) =a tan 2└−sin(β_(p)),−cos(β_(p))sin(α)┘ where ψ is the prescribed yaw attitude (angle), a is the spacecraft orbit angle (i.e. its position), atan2 is the two-argument arctangent function and $\beta_{p} = {\beta_{a} + {\left( \frac{{\alpha } - \frac{\pi}{2}}{\frac{\pi}{2}} \right)^{n}\left\lbrack {{{sign}\quad\left( \beta_{a} \right)\beta_{\min}} - \beta_{a}} \right\rbrack}}$ where β_(a) is the actual current sun beta angle (i.e. the angle between the orbit plane and the sun), β_(min) is a minimum sun beta angle below which the satellite cannot perfectly follow the ideal profile, and the orbit angle a is normalized (“unwrapped”) so that its value always falls between −π and +π. The exponent n is the parameter of the above equations, and it would be adjusted on each pass of the blocks 1008-1012 to achieve the best performance. Note that for this example there are some constraints on n, such as it must be even and positive. While other forms of the candidate profile may yield better performance, the above technique and candidate profile yields a value of n=10.

Conclusion

This concludes the description of the preferred embodiments of the present invention. The foregoing description of the preferred embodiment of the invention has been presented for the purposes of illustration and description. It is not intended to be exhaustive or to limit the invention to the precise form disclosed. Many modifications and variations are possible in light of the above teaching. It is intended that the scope of the invention be limited not by this detailed description, but rather by the claims appended hereto. The above specification, examples and data provide a complete description of the manufacture and use of the composition of the invention. Since many embodiments of the invention can be made without departing from the spirit and scope of the invention, the invention resides in the claims hereinafter appended. 

1. In a system of navigational satellites, at least one of the navigational satellites being controlled by a terestrially-based command station and having a wide beam antenna transmitting a first navigation signal and a spot beam antenna transmitting a second navigation signal, a method of estimating a lever arm correction between the wide beam antenna and the spot beam antenna, comprising the steps of: generating a predicted satellite attitude over the time period T from a predicted position of the at least one satellite over the time period T and a profile of a prescribed satellite attitude maneuver over the time period T; generating predicted gimbal angles of the spot beam antenna over the time period T from the predicted satellite attitude over the time period T and a territorial location of the spot beam when illuminating a navigational receiver over the time period T; predicting the lever arm correction between the wide beam antenna and the spot beam antenna over the time period T from the predicted gimbal angles of the spot beam antenna over the time period T and the predicted satellite attitude over the time period T; and transmitting the predicted lever arm correction from the command station to the navigational receiver via the at least one navigational satellite.
 2. The method of claim 1 wherein the navigational satellite comprises a solar panel and a satellite bus, and wherein the prescribed satellite attitude maneuver is performed to direct a yaw axis of the satellite bus at the Earth while directing the satellite solar panel towards the Sun.
 3. The method of claim 1, wherein the predicted lever arm correction is transmitted from the satellite to the navigational receiver via the first navigational signal or the second navigation signal.
 4. The method of claim 1, wherein the step of predicting the lever arm correction between the wide beam antenna and the spot beam antenna over the rime period T from the predicted gimbal angles of the spot beam antenna over the time period T and the predicted satellite attitude over the time period T comprises the steps of; computing a dine-varying lever arm correction vector; and curve fitting the computed timevarying transformed lever arm correction vector.
 5. The method of claim 1, wherein the predicted satellite attitude is generated from an angle between an orbital plane of the at least one satellite to the Sun and the predicted position of the at least one satellite within the orbital plane.
 6. The method of claim 1, wherein the prescribed satellite maneuver is constrained to limit an attitude rate of the navigational satellite to a maximum value.
 7. The method of claim 1, wherein the prescribed satellite maneuver is constrained to limit the yaw rate of the navigational satellite to a maximum value.
 8. The method of claim 1, wherei the prescribed satellite maneuver is constrained to limit an error between the predicted satellite attitude and the actual satellite attitude to less than 0.2 degrees.
 9. The method of claim 1, wherein the prescribed satellite maneuver is constrained so that a yaw axis of the satellite is directed at the Earth while concurrently directing a satellite solar panel toward the Sun.
 10. The method of claim 1, wherein the predicted gimbal angles of the spot beam antenna over time T are fitter generated from spot beam antenna geometry.
 11. The method of claim 1, wherein the terrestrial location of the spot beam changes during the time period T.
 12. A command station for communicating with a navigational satellite, the navigational satellite having navigational satellite bus, a first satellite antenna for transmitting a first navigational signal to a navigational receiver and a second satellite antenna for transmitting a second navigational signal to the navigational receiver, the command station comprising a lever arm correction module, for predicting a lever arm correction between the first satellite antenna and the second satellite antenna, the lever arm correction module comprising a satellite position predictor module, for generating a predicted navigational satellite bus position over the time period T; a satellite attitude predictor module, for generating a predicted navigational satellite bus attitude over the time period T from the predicted navigational satellite bus position over the time period T and a profile of a prescribed navigational satellite bus attitude maneuver over the time period T; a lever arm prediction module, for predicting the lever an correction between the first antenna and the second antenna over the time period T from a terrestrial location of a spot beat of the second satellite antenna when illuminating a navigational receiver over the time period T, and the predicted navigational satellite bus attitude over the time period T, and a geometry between the navigational satellite bus and the second satellite antenna; and a transmitter for transmitting the predicted lever arm correction to the navigational receiver via the navigational satellite.
 13. The apparatus of claim 12, wherein the navigational satellite comprises a solar panel and a satellite bus, and wherein the prescribed satellite attitude maneuver is performed to direct a yaw axis of the satellite bus at the Earth while directing the solar panel towards the Sun.
 14. The apparatus of claim 12, wherein the predicted lever arm correction is transmitted from the satellite to the navigational receiver via the fist navigation signal or the second navigational signal.
 15. The apparatus of claim 12, wherein the prescribed satellite attitude maneuver is generated from an angle between an orbital plane of the at least one satellite and the Sun and the predicted position of the at least one satellite within the orbital plane and is constrained to limit the yaw rate of the navigational satellite to a maximum value.
 16. The apparatus of claim 12, wherein satellite is commanded to follow the prescribed satellite attitude maneuver, and the prescribed satellite attitude maneuver is constrained to limit an error between the commanded satellite attitude and the actual satellite attitude to less than 0.2 degrees.
 17. The apparatus of claim 12, wherein the lever arm prediction module comprises: a curve fitting module for curve fitting lever arm correction to produce lever arm correction coefficients.
 18. In a system of navigational satellites, at least one of the navigational satellites being controlled by a terrestrially-based command station and having a wide beam antenna transmitting a first navigation signal and a spot beam antenna transmitting a second navigation signal, an apparatus for estimating a lever arm correction between the wide beam antenna and the spot beam antenna, comprising: means for generating a predicted satellite attitude over the time period T from a predicted position of the at least one satellite over the time period T and a profile of a prescribed satellite attitude maneuver over the time period T; means for generating predicted gimbal angles of the spot beam antenna over the time period T from the predicted satellite attitude over the rime period T and a terrestrial location of the spot beam when illuminating a navigational receiver over the time period T; means for predicting he lever arm correction between the wide beam antenna and the spot beam antenna over the time period T from the predicted gimbal angles of the spot beam antenna over the time period T and the predicted satellite attitude over the time period T; and means for transmitting the predicted lever arm correction from the command station to the navigational receiver via the at least one navigational satellite.
 19. The apparatus of claim 18, wherein the navigational satellite comprises a solar panel and a satellite bus, and wherein the prescribed satellite attitude maneuver is performed to direct a yaw axis of the satellite bus at the Earth while directing the satellite solar panel towards the Sun.
 20. The apparatus of claim 18, wherein the predicted lever arm correction is transmitted from the satellite to the navigational receiver via the first navigational signal or the second navigation signal.
 21. The apparatus of claim 18, wherein the means for predicting the lever arm correction between the wide bean antenna and the spot beam antenna over the time period T from the predicted gimbal angles of the spot beam antenna over the the period T and the predicted satellite attitude over the time period T comprises: means for computing a time-varying lever arm correction vector; and means for curve fitting the computed time-via transformed lever arm correction vector.
 22. The apparatus of claim 18, wherein the predicted satellite attitude is generated from an angle between an orbital plane of the at least one satellite and the Sun and the predicted position of the at least one satellite within the orbital plane.
 23. The apparatus of claim 18, wherein the prescribed satellite maneuver is constrained to limit an attitude rate of the navigational satellite to a maximum value.
 24. The apparatus of claim 18, wherein the prescribed satellite maneuver is constrained to limit the yaw rate of the navigational satellite to a maximum value.
 25. The apparatus of claim 18, wherein the prescribed satellite maneuver is constrained to limit an error between the predicted satellite attitude and the actual satellite attitude to less than 0.2 degrees.
 26. The apparatus of claim 18, wherein the prescribed satellite maneuver is constrained so that a yaw axis of the satellite is directed at the Earth while concurrently directing a satellite solar panel toward the Sun.
 27. The apparatus of claim 18, wherein the predicted gimbal angles of the spot beam antenna over time T are further generated from spot beam antenna geometry.
 28. The apparatus of claim 18, wherein the terrestrial location of the spot beam changes during the time period T. 